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A181855 Numerator of Nemes numbers G_n.

%I A181855 #24 Feb 10 2025 23:59:53
%S A181855 1,1,1,239,-46409,9113897,-695818219549,5649766313929,
%T A181855 -1070083202835456443,93856597276403726428217,
%U A181855 -4815785492460413153189484781,674781102986061046417681986493,-9845646538265462155478818981872958283
%N A181855 Numerator of Nemes numbers G_n.
%C A181855 G(n) = A181855(n)/A181856(n). Nemes numbers provide the coefficients for an asymptotic expansion for the Gamma function for real arguments greater than or equal to one.
%C A181855 Gamma(x) = sqrt(2*Pi/x)*((x/e)*(Sum_{k=0..n-1} G_k x^(-2k) + R_n(x)))^x.
%H A181855 Gergő Nemes, <a href="http://dx.doi.org/10.1007/s00013-010-0146-9">New asymptotic expansion for the Gamma function</a>, Arch. Math. 95 (2010), 161-169, Springer Basel.
%H A181855 W. Wang, <a href="http://dx.doi.org/10.1016/j.jnt.2015.12.016">Unified approaches to the approximations of the gamma function</a>, J. Number Theory (2016).
%F A181855 G_0 = 1 and for n > 1 and B_n denoting the Bernoulli number,
%F A181855 G_n = (1/(2*n)) * Sum_{m=0..n} B_{2*m+2} * G_{n-m-1} / (2*m+1).
%F A181855 a(n) = numerator(p(2*n)) with p(n) = Y_{n}(0, z_2, z_3, ..., z_n)/n! with z_k = (k-2)!*Bernoulli(k,1) and Y_{n} the complete Bell polynomials. - _Peter Luschny_, Oct 03 2016
%e A181855 G_0 = 1, G_1 = 1/12, G_2 = 1/1440, G_3 = 239/362880.
%p A181855 G := proc(n) option remember; local k; `if`(n=0,1,
%p A181855 add(bernoulli(2*m+2)*G(n-m-1)/(2*m+1),m=0..n-1)/(2*n)) end;
%p A181855 a181855 := n -> numer(G(n));
%p A181855 # Alternatively:
%p A181855 p := n -> CompleteBellB(n, 0, seq((k-2)!*bernoulli(k, 1), k=2..n))/n!:
%p A181855 a := n -> numer(p(2*n)): seq(a(n), n=0..12); # _Peter Luschny_, Oct 03 2016
%t A181855 a[0] = 1; a[n_] := a[n] = Sum[ BernoulliB[2m + 2]*a[n - m - 1]/(2m + 1), {m, 0, n}]/(2n); Table[a[n] // Numerator, {n, 0, 12}] (* _Jean-François Alcover_, Jul 26 2013 *)
%t A181855 CompleteBellB[n_, zz_] := Sum[BellY[n, k, zz[[1 ;; n-k+1]]], {k, 1, n}];
%t A181855 p[n_] := CompleteBellB[n, Join[{0}, Table[(k-2)! BernoulliB[k, 1], {k, 2, n}]]]/n!;
%t A181855 a[n_] := Numerator[p[2n]];
%t A181855 Table[a[n], {n, 0, 12}] (* _Jean-François Alcover_, Sep 09 2018 *)
%Y A181855 Cf. A000367, A002445, A181856 (denominators).
%K A181855 sign,frac
%O A181855 0,4
%A A181855 _Peter Luschny_, Dec 02 2010